This question is not a duplicate of Why didn't Dumbledore prevent the prosecution of Sirius Black?.
I want to know what evidence Dumbledore had that Sirius wasn't lying in Harry Potter and the Prisoner of Azkaban. It can't be Leglimancy because Sirius might have used Occlumency. Dumbledore wouldn't risk Harry and Hermione's life (there was a werewolf on the loose) based solely on a hunch. So Sirius must have given some concrete evidence. Remember that Dumbledore couldn't have spoken to Remus either. So why would he believe in the Animagus version of what happened when Sirius supposedly blew up the whole street? As far as I am aware, this isn't mentioned fully in the books.
Here are a few explanations:
Dumbledore realized that the whole story fitted together. It explained why Sirius had entered the castle but did not kill Harry. It explained why Sirius was in Ron's bed and not on Harry's. But there was the possibility that Sirius was mad.
He also realized why Buckbeak had disappeared earlier that night. But this can't be a sole reason for thinking some one is innocent. Maybe he used this in connection with (1).
So why did Dumbledore believe Sirius' version of events?